Namely, each \(\Phi \)  is a Banach space, which is an order ideal of \(L^0:=L^0(\mathbb {R}; \mathrm {d}t)\) , that is, for every \(f\in L^0\)  and \(g\in \Phi \)  if \(|f|\le |g|\) , then \(f\in \Phi \)  and \(\Vert f\Vert _\Phi \le \Vert g\Vert \) .
Here, \(L^0\)  stands for the space of all complex measurable functions on \(\mathbb {R}\)  (as usual, any two functions equal almost everywhere are identified).
Moreover, \(\Phi \)  has Fatou's property, and by the Lorentz-Luxemburg theorem [1]}, \((\Phi ^{\prime })^{\prime } = \Phi \)  with equal norms. Here, \(\Phi ^{\prime }\)  denotes the (Köthe) dual (or associated space) of \(\Phi \) ; see [1]}.
